cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A309619 a(n) = Sum_{k=0..floor(n/2)} k! * (n - 2*k)!.

Original entry on oeis.org

1, 1, 3, 7, 28, 128, 754, 5178, 41124, 368220, 3670872, 40290744, 482716896, 6267697920, 87664818960, 1313983544400, 21010949076960, 357007805477280, 6423473819220480, 122003441554176000, 2439346762501367040, 51213306647556506880, 1126446562222595147520
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 10 2019

Keywords

Links

Crossrefs

Cf. A000142, A003149, A096161, A107713, A309618, A339515.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[Sum[k!*x^k, {k, 0, nmax}]*Sum[k!*x^(2*k), {k, 0, nmax}], {x, 0, nmax}], x]
Table[Sum[k!*(n-2*k)!, {k, 0, Floor[n/2]}], {n, 0, 25}]
  • PARI
    a(n) = sum(k=0, n\2, k! * (n - 2*k)!); \\ Michel Marcus, Dec 08 2020

Formula

G.f.: B(x)*B(x^2), where B(x) is g.f. of A000142.
a(n) ~ n! * (1 + 1/n^2 + 1/n^3 + 3/n^4 + 13/n^5 + 57/n^6 + 271/n^7 + 1467/n^8 + 8905/n^9 + 58965/n^10 + ...), for coefficients see A326984.

A110467 Convolution of 4^n*n! and n!.

Original entry on oeis.org

1, 5, 38, 430, 6640, 130200, 3088560, 85828080, 2731899456, 97956720000, 3906077932800, 171436911264000, 8211994618982400, 426284974571904000, 23836815193556736000, 1428394963614554880000, 91316330157374106624000
Offset: 0

Views

Author

Paul Barry, Jul 21 2005

Keywords

Links

Crossrefs

Cf. A107713, A108953.

Programs

  • Mathematica
    Table[Sum[k!*4^k*(n - k)!, {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Aug 28 2017 *)
  • PARI
    for(n=0,50, print1(sum(k=0,n, k!*4^k*(n-k)!), ", ")) \\ G. C. Greubel, Aug 28 2017

Formula

E.g.f. (for offset 1): log((1-x)*(1-4*x))/(4*x-5).
a(n) = n!*Sum_{k=0..n} 4^k/binomial(n, k).
a(n) = Sum_{k=0..n} k!*4^k*(n-k)!.
a(n) ~ 4^n * n! * (1 + 1/(4*n) + 1/(8*n^2) + 7/(32*n^3) + 1/(2*n^4) + 187/(128*n^5) + 1337/(256*n^6) + 22559/(1024*n^7) + 109517/(1024*n^8) + 1202047/(2048*n^9) + 14710847/(4096*n^10) + ...). - Vaclav Kotesovec, Dec 07 2020

A108953 Convolution of 3^n*n! and n!.

Original entry on oeis.org

1, 4, 23, 192, 2184, 31728, 560412, 11630592, 276921216, 7433925120, 222038547840, 7301712936960, 262112637864960, 10198096116526080, 427456901317420800, 19202256562264473600, 920321900537337446400, 46874495077202077286400, 2528269620326135923507200
Offset: 0

Views

Author

Paul Barry, Jul 21 2005

Keywords

Crossrefs

Cf. A107713, A110467.

Programs

  • Mathematica
    Rest[Range[0, 20]! CoefficientList[Series[((Log[1 - 4 x + 3 x^2]))/(3 x - 4), {x, 0, 20}], x]] (* Vincenzo Librandi, Jul 13 2015 *)

Formula

E.g.f. (for offset 1): log(1-4x+3x^2)/((3x-4)).
a(n) = n!*Sum_{k=0..n} 3^k/binomial(n, k).
a(n) = Sum_{k=0..n} k!*3^k*(n-k)!.
a(n) ~ 3^n * n! * (1 + 1/(3*n) + 2/(9*n^2) + 4/(9*n^3) + 32/(27*n^4) + 328/(81*n^5) + 152/(9*n^6) + 20168/(243*n^7) + 341944/(729*n^8) + 2183512/(729*n^9) + 15540472/(729*n^10) + ...). - Vaclav Kotesovec, Dec 07 2020

Extensions

More terms from Vincenzo Librandi, Jul 13 2015

A309618 a(n) = Sum_{k=0..floor(n/2)} k! * 2^k * (n - 2*k)!.

Original entry on oeis.org

1, 1, 4, 8, 36, 140, 832, 5376, 42432, 374592, 3720960, 40694784, 486679296, 6310114560, 88168366080, 1320468480000, 21101183631360, 358354687426560, 6444941507297280, 122367252835860480, 2445878526994022400, 51337143210820239360, 1128918790687649955840
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 10 2019

Keywords

Links

Crossrefs

Cf. A000142, A003149, A107713, A110143, A309619.

Programs

  • Mathematica
    Table[Sum[k!*2^k*(n-2*k)!, {k, 0, Floor[n/2]}], {n, 0, 25}]
nmax = 25; CoefficientList[Series[Sum[k!*x^k, {k, 0, nmax}] * Sum[k!*2^k*x^(2 k), {k, 0, nmax}], {x, 0, nmax}], x]

Formula

G.f.: B(x)*B(2*x^2), where B(x) is g.f. of A000142.
a(n) ~ n! * (1 + 2/n^2 + 2/n^3 + 10/n^4 + 50/n^5 + 250/n^6 + 1442/n^7 + 9514/n^8 + 68882/n^9 + 539098/n^10 + ...), for coefficients see A326983.

A109315 Numbers n such that prime(n) - n is a prime power.

Original entry on oeis.org

12, 15, 38, 39, 118, 152, 190, 258, 462, 690, 746, 1396, 1632, 2119, 3370, 4522, 4600, 7520, 15006, 24222, 33156, 34038, 51372, 52342, 64638, 77470, 90790, 101946, 104670, 156772, 166822, 167700, 175818, 194092, 200022, 229630, 246208, 328462, 362440, 372882
Offset: 1

Views

Author

Zak Seidov and Robert G. Wilson v, May 22 2005

Keywords

Examples

			690 is OK because prime(690)-690 = 5179-690 = 4489 = 67^2, 67 is prime.

Links

Crossrefs

Cf. A025475 = powers of a prime but not prime, also nonprime n such that sigma(n)*phi(n)>(n-1)2; A107712 = values of q, A107713 = values of k; A107714 = values of prime(A109315(n)).
Cf. A083240.

Programs

  • Mathematica
    lst = {}; fQ[n_] := Block[{pf=FactorInteger[n]}, (2-Length[pf])(pf[[1, 2]]-1) > 0]; Do[ If[ fQ[Prime[n] - n], Print[n]; AppendTo[lst, n]], {n, 3, 362439}]; lst

Formula

prime(n) - n = q^k, q is prime and k_Integer >= 2.

A300519 Convolution of n! and n^n.

Original entry on oeis.org

1, 2, 7, 39, 321, 3603, 51391, 884873, 17770445, 406673247, 10431884283, 296262164637, 9224841015745, 312441152401067, 11434829066996087, 449675059390576257, 18908960744072894325, 846638474386244188311, 40213487658138717885907, 2019543479160709325145893
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 08 2018

Keywords

Crossrefs

Cf. A000142, A000312, A061711, A277506.
Cf. A107713, A108953, A110467.

Programs

  • Mathematica
    Table[Sum[If[k == 0, 1, k^k] * (n-k)!, {k, 0, n}], {n, 0, 20}]

Formula

a(n) = Sum_{k=0..n} k^k * (n-k)!.
a(n) ~ n^n * (1 + exp(-1)/n).
Showing 1-6 of 6 results.

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